Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{n^2 + n - 6}{n^2 - 9n + 14}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{n^2 + n - 6}{n^2 - 9n + 14} = \dfrac{(n + 3)(n - 2)}{(n - 7)(n - 2)} $ Notice that the term $(n - 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(n - 2)$ gives: $x = \dfrac{n + 3}{n - 7}$ Since we divided by $(n - 2)$, $n \neq 2$. $x = \dfrac{n + 3}{n - 7}; \space n \neq 2$